Date: Dec 28, 2012 10:58 AM
Author: Milo Gardner
Subject: Re: Archimedes (and Ahmes) square root of 5, 6 and 7

Peter,

my work contains typos .. the problem you cited was not in context. a quotient 5 with remainder 1/4 was not possible on a STEP 1.

had a quotient 5 appeared ... I.E

ESTIMATING the square root of 29.

step 1 would have found R = (29-25)/2Q = 4/10 = 2/5

(5 + 2/5)^2 = 29 + 4/25

STEP 2

reduce error 4/25 by dividing by 2( 5 + 2/5)

4/25 x 5/54 = 2/27

hence a final unit fraction series would replace

(5 + 2/5 - 2/27)^2 by considering

(5 + 1/5 + (27-10)/135)^2 = (5 + 1/5 + 1/9 + 2/135)^2

was accurate to (2/27)^2

NOTE: the conversion of 2/135 to a unit fraction series follows Ahmes 2/n table rules

(2/5)(1/27) = (1/3 + 1/15)(1/27) = 1/51 + 1/405

MEANT THE SQUARE ROOT OF 29 WAS ESTIMATED IN TWO STEPS BY

(5 + 1/5 + 1/9 + 1/51 + 1/405)^2